Preliminary Examination: Classical Mechanics Department of Physics and Astronomy University of New Mexico Spring 2010

Size: px

Start display at page:

Download "Preliminary Examination: Classical Mechanics Department of Physics and Astronomy University of New Mexico Spring 2010"

Dustin Norris
5 years ago
Views:

1 Preliminary Examination: Classical Mechanics Department of Physics and Astronomy University of New Mexico Spring 2010 Instructions: The exam consists of 10 problems (10 points each). Where possible, show all work; partial credit will be given if merited. Personal notes on two sides of an 811 page are allowed. Total time: 3 hours. Unless otherwise noted, commonly used symbols are de ned as follows: _x : Rate of change of position x with time. x : Rate of change of velocity _x with time. t : Time x(0) : Initial position _x(0) : Initial velocity L : Lagrangian r : Radius : Polar angle in xy plane : Azimuthal angle ^x : Unit vector in the x direction. 1

2 P1. A geostationary orbit (or Geostationary Earth Orbit - GEO) is an orbit directly above the Earth s equator (0 latitude), with a period equal to the Earth s rotational period and an orbital eccentricity of approximately zero. Geostationary objects appear motionless in the sky from points on the surface of the earth, making the GEO an orbit of great interest to operators of communications and weather satellites. Calculate the radius of a GEO satellite. Express your answer as a multiple of the Earth s radius R E. 2

3 P2. A photon with initial momentum p collides with a free electron having a mass m that is initially at rest. If the electron and photon recoil in opposite directions, what will be the change in the photon s wavelength? (Hint: use relativistic forms for energy and momentum.) 3

4 P3. The relative velocity u between a rocket and its ejected exhaust gas is a constant. Consider launching a rocket in the vertical direction. If the initial mass of the rocket is M 0 ; at what minimum rate must exhaust mass be ejected so that the rocket will experience an upward acceleration? 4

5 P4. A particle of mass m moves in three dimensions. A Lagrangian governing its dynamics is given by L = 1 r 2 m 2 2 _ + r 2 2 _ sin 2 mgr cos where r; ; and are the radius, azimuthal angle, and polar angle, in spherical coordinates, and g is a constant. Write down the associated equations of motion. Describe and sketch a physical system for which this is an appropriate Lagrangian. 5

6 P5. Consider a single particle with mass m that is con ned to a rectangular container having sides of length L x ; L y ; and L z : The particle moves with an initial velocity ~v 0 = hv 0x ; v 0y ; v 0z i : The vector components will change in sign, but not in magnitude, because collisions with the walls are elastic. A transducer records the force F x (t) exerted on the right hand face whenever the particle makes a collision. A sketch of F x versus t is shown in the gure below. Calculate the average force on the right hand face, and determine the pressure. 6

P6. The picture shows a rotating platform that serves as a playground merry go round. The platform rotates on low-friction bearings about its center axis. It has a radius of 2.

7 P6. The picture shows a rotating platform that serves as a playground merry go round. The platform rotates on low-friction bearings about its center axis. It has a radius of 2.0 meters, and its moment of inertia about the center axis is 200 kgm 2 : Suppose that the platform is given an initial rotation rate of 1.0 radians per second. A small dense brick having a mass of 50 kg, initially at rest, is then placed on the platform at a distance of 0.5 meters from the rotation axis. The brick initially slides, but eventually stops sliding at a distance 1.0 m from the axis. How many Joules of mechanical energy are converted to heat? 7

8 P7. Consider a planet of mass m orbiting the sun with mass M: In the plane of the orbit, the planet s motion is described by the two coupled equations, m d2 r dt 2 = mr 2 d dt GMm r 2 + `2 mr 3 ; (1) = `; (2) where r; are polar coordinates locating the planet with respect to the sun, G is the gravitational constant, and ` is a constant. With the aid of eq. (2) one can write d2 r ` 2 dt = 2 m u 2 d 2 u; where u () = 1=r () describes the trajectory in d 2 the orbital plane. Using this result, show that eq. (1) can be written as, `2 d 2 u m d 2 = `2 GMm + m u: Noting that `; m; M, and G are all constants, solve this equation explicitly and show that this describes a closed orbit. 8

9 P8. An electron having mass m and charge e experiences viscous damping as it moves along the x axis under the in uence of an ac electric eld. Its motion is described by a time-dependent Lagrangian 1 L = e t 2 m ee _x _x2 +! 2 ( cos!t +! sin!t) ; + 2 where ;!; and E are positive constants. For initial conditions x(0) = _x(0) = 0; nd at least one constant of the motion, and obtain an expression for _x(t) that describes the behavior at long times. 9

10 P9. Two particles having the same mass m hang in series from two springs having the same sti ness constant k; as shown in the gure. What are the frequencies of the two normal modes of vibration? 10

11 P10. A sti wire hoop of radius R is xed in space, and there is no gravity. A small bead with mass m slides along the hoop. The coe cient of sliding friction between the bead and the wire is 0:10: If the bead is given an initial angular velocity of 10 revolutions/second, how long does it take for the bead to coast around the hoop 10 times? 11

PHYS 1303 Final Exam Example Questions

PHYS 1303 Final Exam Example Questions (In summer 2014 we have not covered questions 30-35,40,41) 1.Which quantity can be converted from the English system to the metric system by the conversion factor